Solve Math Word Problem: Ant Spiraling Journey

In summary, the ant starts at the origin (0,0) and walks in a spiral path by moving one unit right, one-half unit up, one-quarter unit left, one-eighth unit down, etc. It always turns counter-clockwise at a 90 degree angle and goes half the distance it went on the previous move. The goal is to find the point (x,y) where the ant is approaching on its journey. The problem can be solved by breaking down the x and y displacements into geometric series and finding their sums, which are simple geometric series since they alternate between positive and negative values. The ant's journey can be described as chaotic, but can be solved using mathematical principles.
  • #1
Macleef
30
0

Homework Statement



An ant of negligible dimensions start at the origin (0,0) of the standard 2-dimensional rectangular coordinate system. The ant walks one unit right, then one-half unit up, then one-quarter unit left, then one-eighth unit down, etc. In each move, it always turn counter-clockwise at a 90 degree angle and goes half the distance it went on the previous move. Which point (x,y) in the xy-plane in the ant approaching in its spiraling journey?

Homework Equations



I think you use the geometric series to solve this problem?


The Attempt at a Solution



I don't have an attempt at this problem because I don't know where to begin!
I don't know how to solve this problem! All I know is you use the geometric series??
And if you do, how would you go solve this problem with the geometric series?


The answer is: (4/5 , 2/5)
 
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  • #2
Write down a series of all of the x displacements and another series of all the y displacements. They should be geometric series. Then you can start worrying about summing them.
 
  • #3
The sum will be unto infinity.
 
  • #4
It appears to be ChaosEverlasting's goal to spread everlasting chaos!

As Dick suggested, look at "x" (East,West) and "y" (North, South) components separately. That's easy since the ant alternates between going East-West and North-South.

Yes, as ChaosEverlasting implies, you will have two infinite series. However, since they are alternating series (positive, then negative), both series converge. In fact, they are simple geometric series.
 
  • #5
Do you know how to find the sum of an infinite geometric series?
 
Last edited:
  • #6
If I was an Ant, I'd start swinging at you. "Negligible Dimensions", pfft. =]
 
  • #7
Yeah, ants tend to be really sensitive about their size!
 

1. What is the Ant Spiraling Journey math problem?

The Ant Spiraling Journey is a math word problem that involves a scenario where an ant starts at the center of a circular table and crawls along the edge of the table in a spiral shape until it reaches the center again.

2. What information is needed to solve the Ant Spiraling Journey problem?

To solve this problem, you will need to know the diameter of the circular table and the length of each spiral the ant crawls. You will also need to know the total distance traveled by the ant.

3. How is the Ant Spiraling Journey problem solved?

The problem can be solved using the formula for finding the circumference of a circle, which is C = πd (where C is the circumference and d is the diameter). First, find the total distance traveled by the ant by multiplying the number of spirals by the length of each spiral. Then, divide this distance by the circumference of the table to find the number of times the ant goes around the table. Finally, divide the number of times the ant goes around the table by 4 to find the number of complete rotations.

4. Can the Ant Spiraling Journey problem be solved using different units of measurement?

Yes, the problem can be solved using any unit of measurement as long as the units are consistent. For example, if the diameter of the table is in inches, the length of each spiral should also be in inches.

5. What real-life applications can be related to the Ant Spiraling Journey problem?

This problem can be related to real-life scenarios such as finding the distance a car travels in a circular racetrack or the number of rotations a fan blade makes in a certain amount of time. It also helps develop spatial awareness and critical thinking skills.

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